A graph is nonseparable if it has no cutpoints; a block is a maximal nonseparable subgraph [1]} (p. 26). A family of subsets of some given set determines
an intersection graph where vertices correspond to subsets and adjacency
holds if and only if the corresponding subsets (are distinct and) have nonempty intersection.
The intersection graph formed by the cutpoint singletons and the vertex sets
of the blocks is a forest, called the block-cutpoint forest [1]} (p. 36). The
points of the intersection graph which have degree 1 must be blocks since
every cutpoint belongs to more than one block. A graph is connected if and
only if its block-cutpoint forest is a tree.
